١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

Ira N. Levine, Quantum Chemistry + MOLECULAR ORBITALS FOR H2 EXCITED STATES

we construct approximate functions for further excited + states so as to build up a supply of H2 -like molecular orbitals. We shall then use these MOs to discuss many- diatomic molecules qualitatively, (just as hydrogenlike AOs).

we can use the linear-variation-function method:

we add in more AOs to the previous linear combination.

+ for the six lowest H2 σ states,

three lowest m = 0 hydrogenlike functions

For the symmetry of the homonuclear diatomic molecule

+ : even (g) states _ : odd (u) states

١ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

The relative magnitudes of the coefficients:

For the two states that dissociate into a 1s hydrogen :

c1 >> c2 and c3 (c2 and c3 vanish in the limit of R going to infinity)

As a first approximation,

LCAO functions

From the viewpoint of perturbation theory: if we take the separated as the unperturbed problem, the above functions are the correct zeroth-order wave functions.

for the two states that dissociate to a 2s hydrogen atom

c2 >> c1 and c3 (c1 and c3 vanish in the limit of R going to infinity)

As a first approximation,

LCAO functions To find rigorous upper bounds to the energies of these + two H2 states, we must use the upper trial function and solve the appropriate secular equation

contour diagrams

٢ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

σ MOs even though they correlate with 2p separated AOs, since they have m = 0.

dashed lines ≡ nodal surfaces

For the hydrogen atom, the 2s and 2p AOs are degenerate, and so …

+ In the R->  limit, H2 consists of an H atom perturbed by the essentially uniform electric field of a far-distant proton.

for the n = 2 levels

the correct zeroth-order functions

For molecules that dissociate into many-electron atoms, the separated- atoms 2s and 2p AOs are not degenerate but do lie close together in energy. Hence, …

٣ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

For the other two 2p atomic orbitals, we can use

either the 2p+1 and 2p-1 complex functions or the 2px and 2py real functions.

+ inversion of the electron's coordinates in H2

Ali Ebrahimi ۴ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

inversion of the electron's coordinates

a u orbital.

Reflection in the plane perpendicular to the axis and midway between the nuclei:

unchanged, an unstarred (bonding) orbital.

≡

Ali Ebrahimi ۵ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

≡

the λ = 1 energy levels are doubly degenerate, corresponding to m = 1

have the same shapes,

do not give charge buildup between the nuclei

Ali Ebrahimi ۶ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

the more familiar alternatives:

builds up probability density

is not symmetrical nodal plane about z

Bonding (u)

antibonding (g)

and have the same energy. are not eigenfunctions of Lz

linear combinations of are eigenfunctions of Lz

and have the same energy. are not eigenfunctions of Lz

linear combinations of are eigenfunctions of Lz

٧ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

+ how H2 MOs correlate with the united-atom AOs.

R  0

R  0 p united-atom states

R  0 d united-atom states

?

MO CONFIGURATIONS OF HOMONUCLEAR DIATOMIC MOLECULES

If we ignore the interelectronic repulsions, the zeroth- + order wave function is a Slater determinant of H2 -like one-electron spin-orbitals. + We approximate the spatial part of the H2 spin-orbitals by the LCAO-MOs The sizes and energies of the MOs vary with varying internuclear distance for each molecule and vary as we go from one molecule to another.

the order in which the MOs fill as we go across the :

٨ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

Molecular-Orbital Nomenclature for Homonuclear Diatomic Molecules

the noncrossing rule: the energies of MOs with the same symmetry cannot cross. symmetry refers to whether the orbital is g or u and whether it is σ, π, δ....

s, d, g,... united-atom AOs (even) correlate with g MOs, p, l, h,... AOs (odd) correlate with u MOs.

Correlation diagram for homonuclear diatomic MOs. this diagram is not quantitative.

٩ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

Homonuclear diatomic MOs formed from 1s, 2s, and 2p AO's.

Molecular electronic configurations

terms, levels, and states

+ H2

De = 4.75 eV H2

no net bonding, He2

 It has bound excited electronic states,

 about two dozen such bound excited states of He2 have been spectroscopically observed in gas discharge tubes.  such excited states decay to the ground electronic state, and then the molecule dissociates.

+ He2

١٠ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

Li2 a single bond indicate the negligible change in inner- shell orbital energies on molecule formation

no net bonding Be2 .

two net bonding B2 electrons one electron in the MO and the other in the MO,

is at variance with the notion that single bonds are always σ bonds.

the electron-spin-resonance spectrum showed that the ground term is a triplet with S = 1

C2 with four net bonding electrons

the and MOs have nearly the same energy in many molecules.

giving a triplet term.

a singlet term. the ground term by a small margin (0.09 eV)

N2 a triple bond

O2 a double bond.

Spectroscopic evidence indicates that in O2 (and in F2) the σg2p is lower in energy than the πu2p MO. The paramagnetism of O2

١١ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

a single bond. F2

Ne2 no net bonding electrons

Na2

0.02 eV below Al2

0.05 eV below Si2

the harmonic vibrational frequency

١٢ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

Bonding MOs produce charge buildup between the nuclei, whereas antibonding MOs produce charge depletion between the nuclei.

Usually, removal of an electron from: a bonding MO decreases De an antibonding MO increases De

+ N2 → N2 : the dissociation energy decreases + O2 → O2 : the dissociation energy increases

Energy is always required to ionize a stable molecule, no matter which electron is removed.

Are interaction between two ground-state He atoms strictly repulsive ? helium gas can be liquefied !

van der Waals forces: all kinds of intermolecular forces. For example: London or dispersion force

Except for highly polar molecules: the dispersion force is the largest contributor to intermolecular attractions.

van der Waals molecules

Ar2 van der Waals molecules: De = 0.012 eV and Re = 3.76 A; Ar2 has seven bound vibrational levels (v = 0,..., 6).

١٣ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

Examples of diatomic van der Waals molecules:

Re, De

Ne2 : 3.1 Å, 0.0036 eV

HeNe : 3.2 Å, 0.0012 eV

Ca2 : 4.28 Å, 0.13 eV

Mg : 2,3.89 Å, 0.053 eV

ELECTRONIC TERMS OF DIATOMIC MOLECULES For atoms, each set of degenerate atomic orbitals constitutes an atomic subshell. For molecules, each set of degenerate molecular orbitals constitutes a molecular shell.

one MO Two MOs

a molecular electronic configuration  giving the number of electrons in each shell

١۴ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

For a many-electron diatomic molecules:

Lz = MLħ, where ML = 0, ±1, ±2, ± ….

2 electronic energy depends on ML  double degeneracy

The magnitude of S :

S = 0, 1/2, 1, 3/2, …

The component of S along an axis = Msħ, where Ms = S, S - 1,..., -S. 2S + 1 is called the spin

2S+1Λ

Diatomic electronic states that arise from the same and that have the same value for Λ and the same value for S are said to belong to the same electronic term.

١۵ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

for a filled-shell molecular configuration:

Ms = ∑ ms values = 0  S = 0

A filled σ shell : ML = ∑m = 0

A filled π shell : ML = ∑m = (-1) + (+1) = 0

A filled δ, ϕ, … shell : ML = ∑m = 0

a closed-shell molecular configuration: S = 0 , Λ = 0 only a 1∑ term

ground electronic configuration of H2

A single σ electron: s = ½  S = ½ → 2∑ A single π electron: s = ½  S = ½ → 2∏ ….

more than one electron:

Nonequivalent : Electrons that are in different molecular shells Equivalent : Electrons that are in same molecular shells

two nonequivalent electrons:

σσ configuration M = 0 1 3 L S = 1 or 0 ∑ and ∑ Each s = ½ σπ configuration: M = 1 1 3 L S = 1 or 0 ∏ and ∏ Each s = ½

١۶ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

πδ configuration,

π electron: m = M = +3, -3, +1, -1 Λ = 3 or 1 ±1 L δ electron: m = ±2

1∏, 3∏, 1Φ, 3Φ.

ππ configuration, each electron ML has m = ±1 values 2, -2, 0, 0 Λ= 2,0, and 0

1Δ, 3Δ, 1∑, 3∑, 1∑, and 3∑

apart from spin degenerate states nondegenerate degeneracy:

Forms of the wave functions for the terms

two π subshells: π and π' M = +2 L a subscript indicates the m value

For the Δ terms: both electrons have m = +1 or both have m = - 1.

neither symmetric the spatial factor in nor antisymmetric the wave function is with respect to … and are unacceptable

١٧ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

both electrons symmetric having m = + 1 antisymmetric

both electrons symmetric having m = - 1 antisymmetric

symmetric or antisymmetric with respect to exchange of electron

 two states

 six states

For the ∑ terms: one electron with m = +1 and one with m = - 1.

Combining them to get symmetric and antisymmetric functions,

symmetric

antisymmetric

١٨ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

two states

six states

These functions have eigenvalue +1 or -1 with respect to reflection of electronic

coordinates in the xz (σv) symmetry plane containing the molecular (z) axis; the superscripts + and - refer to this eigenvalue.

Previous Δ terms are not eigenfunctions of the symmetry operator

a twofold commutes with the degeneracy Hamiltonian

Ôσv eigenvalues

+ +1 same energy - - 1

are not eigenfunctions of Lz but are superpositions of Lz eigenfunctions with eigenvalues +2 and -2.

Λ-type doubling: a very slight splitting of the two states of a 1Δ term by the interaction between the molecular rotational and the electronic orbital angular momentum

١٩ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

equivalent electrons:

π2 configuration

m1 m2 ms1 ms2 ML MS S ٠ 0 2 1/2 -1/2 +1 +1 ٠ 0 -2 1/2 -1/2 -1 -1 ١ 1 0 1/2 1/2 -1 +1 ١ -1 0 -1/2 -1/2 -1 +1 +1 -1 1/2 -1/2 0 0 1 ٠ 0 0 1/2 -1/2 -1 +1

1 ML = 2, -2  Λ = 2  Δ S = 0

3 ML = 0  Λ = 0  ∑ S = 1 (for Ms = -1, 0, 1)

1 ML = 0  Λ = 0  ∑ S = 0 (for Ms = 0

For homonuclear diatomic molecules, a g or u right subscript is added to the term symbol to show the parity of the electronic states belonging to the term. Terms arising from an electron configuration that has an odd number of electrons in molecular orbitals of odd parity are odd (u); all other terms are even (g).

٢٠ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

2 O2 has a π configuration:

The v = 0 level

energy The v = 0 level

Singlet O2 is a reaction intermediate in many organic, biochemical, and inorganic reactions.

Most stable diatomic molecules: 1∑+ ground term

2 Exceptions: B2, Al2, Si2, and O2, and NO ∏ ground

Spectroscopists prefix the the symbol X ground term of a molecule terms of the same spin are designated as multiplicity as the ground term A, B, C,...,

excited terms of different spin multiplicity are designated as a, b, c

ground terms Exceptions are C2 and N2 But, A, B, C,... are used for excited triplet terms

٢١ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

spin-orbit interaction can split a molecular term into closely spaced energy levels

The projection of the total electronic spin S on the molecular axis is Msħ.

In molecules the Ms is called ∑

total axial component of electronic angular momentum (Λ + ∑)ħ

is written as a right subscript to the term symbol

Λ = 2 and S = 1

The spin-orbit interaction energy in diatomic molecules

spin-orbit interaction energy ≈ AΛ∑ A depends on Λ and R but not on ∑  spacing between levels = cte.

A > 0  the multiplet is regular → level with the lowest Λ + ∑ lies lowest A < 0  the multiplet is inverted Λ≠ 0 → 2S + 1 = the number of multiplet components

with Λ≠ 0, levels are doubly degenerate (for two values of ML).

Spin-orbit interaction splits the 3Δ term into three levels, each doubly degenerate, which is removed by the Λ-type doubling.

For ∑ terms, the spin-orbit interaction is very small (quantum numbers ∑ and Ω are not defined). ∑ terms  single nondegenerate .

٢٢ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

THE HYDROGEN MOLECULE

Interparticle distances in H2.

purely electronic Hamiltonian for H2 is:

prevents the equation from being separable 1 and 2 for electrons a and b for nuclei approximation methods

1) the molecular-orbital approach:

ground-state electronic configuration

approximate wave function:

٢٣ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

σg1s = f omission of the spin factor does not affect the variational integral. we want to choose f so as to minimize:

over spatial coordinates of two electrons.

Ideally, f should be found by an SCF calculation. + For simplicity, we can use an H2 -like MO.

+ if we omit the l/r12  Ĥ = sum of two H2 Hamiltonians

+ a good approximation to the ground-state H2 wave function (last sections):

variation function = the product of two such LCAO functions

+ the effective nuclear charge ζ will differ from k for H2 .

٢۴ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

Coulson, Re = 0.732 Å (true value = 0.741 Å) De = 3.49 eV (true value = 4.75 eV + ζ (at 0.732 Å) = 1.197 < k for H2 . the screening of the nuclei from each electron by the other electron.

Kolos and Roothaan improved the results. They expanded f in elliptic coordinates. f is a function of ξ and η (Since m = 0 for the , the eim = 1)

p and q are integers and α and apq are variational parameters. The Hartree-Fock results: Re = 0.733 Å and De = 3.64 eV, which is not much improvement over the the simple LCAO molecular orbital.

we must go beyond the SCF approximation of writing f(1)f(2).

configuration interaction (CI):

we include contributions from SCF (or other) functions for all the excited states with the same symmetry as the ground state. gs configuration

first exs g  u  odd configuration odd parity

To simplify things, we will use the LCAO-MOs as approximations to the MOs.

c is a variational parameter.

Weinbaum,

Re = 0.757 Å ; De = 4.03 eV ; ζ = 1.19 a considerable improvement over the HF result

٢۵ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

We can improve on this result by using a better form for the MOs of each configuration and by including more configuration functions.

James and Coolidge, use of r12 in H2 trial functions:

variational parameters : δ and the cmnjkp function is symmetric with respect to interchange of electrons 1 and 2 (antisymmetric ground-state spin function).

With 13 terms:

De = 4.72 eV, only 0.03 eV in error.

THE -BOND TREATMENT OF H2

the valence-bond (VB) theory, by Heitler and London

more closely related to the chemist's idea of molecules (atomic cores + bonding)

first step : approximate the molecule as two ground-state hydrogen atoms.

1 and 2 for electrons ; a and b for nuclei

Also:

the trial variation function

٢۶ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

trial variation function

determinantal secular equation

We can also consider the problem using perturbation theory: perturbation two neutral ground-state hydrogen atoms molecule

correct zeroth-order wave functions

secular determinant

Hamiltonian is Hermitian, all functions

are real, f1 and f2 are normalized

٢٧ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

the ground state of H2 is a ∑ state with the antisymmetric spin factor and a symmetric spatial factor. Hence 1 must be the ground state.

The Heitler-London ground-state wave function:

for the three states of the lowest 3∑ term are:

the ground-state energy expression:

The Heitler-London calculation does not introduce an effective nuclear charge into the 1s function. Hence lsa(1) is an eigenfunction of Ĥa(1) with eigenvalue – 1/2 Coulomb integral hartree,

٢٨ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

To obtain the U(R) potential-energy curves, we add the internuclear repulsion 1/R to these expressions.

+ Many of the integrals have been evaluated in the treatment of H2

The only new integrals are those involving l/r12.

two-center, two-electron exchange integral:

Two-center : the integrand contains functions centered on two different nuclei two-electron : the coordinates of two electrons occur in the integrand

This must be evaluated using an expansion for 1/r12 in confocal elliptic coordinates

Heitler-London treatment : De = 3.15 eV, Re = 0.87 Å the experimental values : De = 4.75 eV, Re = 0.741 Å In this treatment, most of the binding energy is provided by the exchange integral A.

٢٩ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

Improvements on the Heitler-London function

Wang introduced an orbital exponent ζ in the 1s function:

ζopt = 1.166 , Re = 3.78 eV , De = 0.744 Å

Rosen: mixing in some 2pz character into the atomic orbitals (hybridization). to improve the Heitler-London-Wang function

allows for the polarization of the AOs on molecule formation (De = 4.04 eV)

Another improvements: 1) the use of ionic structures 2) the generalized valence-bond method

COMPARISON OF THE MO AND VB THEORIES

The H2 ground state:

the spatial factor of the unnormalized LCAO-MO wave function

a an centered on nucleus a simplest treatment:  is a 1s AO

the physical significance of the terms:

ionic terms, covalent terms,

Two terms occur with equal weight,

٣٠ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

We remedy this; the simplest procedure is to omit the ionic terms of the MO function.

As the Heitler-London function

there is some probability of finding both electrons near the same nucleus,

variational parameter

 represents ionic-covalent resonance.  a time-independent mixture of covalent and ionic functions.  () = 0 Weinbaum’s calculation:

(Re) = 0.26;  = 1.19; De = 4.03 eV

 = 0 : VB,imp  VB function ;  = 1 VB,imp  LCAO-MO function

(Re) = 0.26 is closer to zero than to 1, and, in fact, the Heitler-London- Wang VB function gives a better dissociation energy than the LCAO- MO function. compare the imp VB trial function with the simple LCAO-MO function improved by CI.

The LCAO-MO CI trial function (unnormalized):

multiplying it by 1/(1 -)

improved MO function and the improved VB function are identical.

٣١ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

 The MO function underestimates electron correlation (ec can be introduced by CI)  The VB function overestimates electron correlation (ec is reduced by ionic-covalent resonance)  at large R, the simple VB is more reliable than the simple MO method

How VB and MO methods divide the H2 electronic Hamiltonian into unperturbed and perturbation Hamiltonians?

For the MO method,

unperturbed Hamiltonian + (sum of two H2 Hamiltonians)

the zeroth-order MO wave function is a product of two H2+-like wave functions (approximate as LCAOs)

The effect of the l/r12 perturbation is taken into account in an average way through use of self-consistent-field molecular orbitals instantaneous electron correlation can be taken into account by (CI) LCAO-MO SCF CI For the valence-bond method,

or

unperturbed system: two hydrogen atoms We have two zeroth-order functions (belong to a degenerate level) The correct ground-state zeroth-order function is the linear combination MO is computationally much simpler than the VB method. The MO method was developed by Hund, Mulliken, and Lennard-Jones in the late 1920s.

٣٢ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

MO AND VB WAVE FUNCTIONS FOR HOMONUCLEAR DIATOMIC MOLECULES

 The MO approximation puts the electrons of a molecule in molecular orbitals (is approximated by LCAOs) which extend over the whole molecule.  The VB method puts the electrons in AOs and constructs the molecular wave function by allowing for "exchange" of the valence electron pairs between the atomic orbitals of the bonding atoms.

The ground state of He2:

 The separated helium atoms: closed-subshell configuration (1s2).  unpaired electrons = 0  VB wave function: antisymmetrized product of the AO functions.

VB wave function

spin function β.

Ideally: is an SCF atomic function helium-atom 1s Approximatly: a hydrogenlike function with effective nuclear charge.

predicts no bonding

shorthand notation:

٣٣ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

MO approximation to the wave function:

the ground-state configuration:

No bonding is predicted,

MO wave function:

The simplest way to approximate the (unnormalized) MOs is LCAOs:

add column 1 to column 3 and column 2 to column 4

subtract column 3 from column 1 and column 4 from column 2

interchange of columns 1 and 3 and of 2 and 4 multiplies by (-1)2

is identical to the VB function

٣۴ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

the simple VB and simple LCAO-MO methods give the same approximate wave functions for diatomic molecules formed from atoms with completely filled atomic subshells.

trial function:

the variational integral

repulsive curve for the interaction of two gs He atoms

the Heitler-London VB functions (gs) for H2 as Slater determinants:

electron on a is paired with an electron of opposite spin on b (Lewis structure H—H)

٣۵ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

HL functions for the lowest H2 triplet state:

2 Li2:  gs configuration 1s 2s,  the Lewis structure is Li—Li (two 2s Li electrons paired and the 1s electrons remaining in the inner shell of each atom)

the gs VB function:

Exercise: show that it is an eigenfunction of the spin 2 operators Ŝ and Ŝz with eigenvalue zero for each operator,

To save space:

pairing (bonding) of the 2sa and 2sb AOs

MO wave function (gs)

approximate the two

lowest MOs by 1sa ± 1sb

2 Recall the notation KK(σg2s) for gs configuration

٣۶ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

The N2 ground state:

the VB treatment: The lowest configuration of N

The Lewis structure : :N≡N:

the VB wave function? three pairs of orbitals and two ways to give opposite spins. Hence, there are 23 = 8 possible Slater determinants that we can write.

In all eight determinants, the first eight columns will remain unchanged

There are six other determinants

The VB wave function is a linear combination of eight determinants .

Rule in the combination?

The single-determinant gs N2 MO function is easier to handle than the eight-determinant VB function.

٣٧ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

EXCITED STATES OF H2

two substantial minima

The lowest MO configuration (a nondegenerate level)

LCAO-MO functions

triply degenerate

nondegenerate

 one bonding and one antibonding electron, and repulsive p-e curves are expected .  Actually, the B level has a minimum in its U(R) curve.

٣٨ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

The Heitler-London wave functions

 going across the periodic : 2σu MO fills before the two 1πu  in H2: 1πu < 2σu

triplet lying lower

levels Each level is twofold degenerate (eight electronic states )

SCF WAVE FUNCTIONS FOR DIATOMIC MOLECULES • Some examples of SCF MO WFs for diatomic molecules.

• The spatial orbitals φi in an MO WF are each expressed as a linear combination of a set of one-electron basis functions χs

• For SCF calculations on diatomic molecules, one can use STOs centered on the various atoms of the molecule as the basis functions. • Complete set of AO basis functions ≡ an infinite number of STOs • True molecular HF wave function can be closely approximated with a reasonably small number of carefully chosen STOs. • A minimal basis set (MBS) for a molecular SCF calculation consists of a single basis function for each inner-shell AO and each valence- shell AO of each atom. • An extended basis set EBS) is a set that is larger than a minimal set. • MBS SCF calculations are easier than EBS calculations, but the latter are much more accurate.

٣٩ Ali Ebrahimi ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

SCF WAVE FUNCTIONS FOR DIATOMIC MOLECULES

Examples of SCF wave functions for diatomic molecules:

1) the SCF MOs for the ground state of Li2 at R = Re:

 The AO functions in these equations are STOs, except for 2s.  A ST 2s AO has no radial nodes and is not orthogonal to a 1s STO.  The HF 2s AO has one radial node (n - I - 1 = 1) and is orthogonal to the 1s AO.

By Schmidt orthogonalization)

S is the overlap integral <1s|2s>.

2pσ for an AO  the p orbital points along the molecular (z); a 2pz AO The 2px and 2py AOs are called 2pπ AOs

optimum orbital exponents: ζ1s = 2.689, ζ2s = 0.634, ζ2pσ = 0.761

six AOs (as basis functions)  approximations for the six lowest MOs of

ground-state Li2; only three of these MOs are occupied.

previous simple expressions

Ali Ebrahimi ۴٠ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

Comparison: simple expressions:

Approximations are good

substantial 2pσ AO contributions in addition to the 2s AO the 2s and 2p AOs are close in energy; the hybridization allows for the polarization of the 2s AOs in forming the molecule.

2) the 3σg MO of the F2 ground state at Re:

a) using a minimal basis set:

Etotal = -197.877 hartrees

b) using an extended basis set:

Etotal = -198.768 hartrees Eexp = —199.670 hartrees the correlation energy ≈ -0.90 hartrees = - 24.5 eV.

Ali Ebrahimi ۴١ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

• Etot = -197.877Eh and -198.768Eh for the minimal and extended calculations, respectively

• Extrapolation to larger basis sets: EHF (Re) = -198.773Eh

• The experimental energy at Re : U(Re)= -199.672Eh. • The correlation-energy definition: nonrelativistic E of the molecule - EHF.

• The relativistic contribution to E of F2 calculated: -0.142Eh, • exact nonrelativistic E at Re : -199.672Eh + 0.142Eh = -199.530Eh.

• Therefore, the correlation energy in F2 is -199.530Eh + 198.773Eh = - 0.757Eh = -20.6 eV.

one needs more than one STO of a given n and l in the linear combination of STOs that is to accurately represent the Hartree-Fock MO.

STOs with different orbital exponents

AOs with quantum number m = 0, that is, the 3d0 and 4f0 AOs

The SCF calculations make clear that all MOs are hybridized to some extent. Thus any diatomic-molecule σ MO is a linear combination of 1s,

2s, 2p0, 3s, 3p0,3d0,... AOs of the separated atoms.

Ali Ebrahimi ۴٢ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

which AOs contribute to a given diatomic MO? 1) only σ--type AOs (s, pσ, dσ,...) can contribute to a σ MO; only π- type AOs (pπ, dπ,...) can contribute to a π MO; and so on. 2) only AOs of reasonably similar energy contribute substantially to a given MO.

Hartree-Fock MO electron-density contours for the ground electronic state

of Li2 as calculated by Wahl.

Hartree-Fock wave functions are only approximations to the true wave functions.

 a HF wave function gives a very good approximation to the electron probability density ρ(x,y, z) for equilibrium configuration.  A molecular property that involves only one-electron operators can be expressed as an integral involving ρ. Consequently, such properties are accurately calculated using HF wave functions (For example, the molecular dipole moment). LiH : with a near HF : dipole moment = 6.00 D (experimental value = 5.83 D) NaCl: the calculated value = 9.18 D (experimental value 9.02 D)  An error of about 0.2 D is typical in such calculations, but where the dipole moment is small, the percent error can be large. CO: (experimental moment = 0.11 D with the polarity C-O+, the near-HF moment = 0.27 D with the wrong polarity C+0-. a CI wave function gives 0.12 D with the correct polarity

Ali Ebrahimi ۴٣ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

A major weakness of the Hartree-Fock method is its failure to give accurate molecular dissociation energies.

N2: HF De = 5.3 eV by an extended-bs (true value = 9.9 eV)

F2: HF De = -1.4 eV (true De = 1.66 eV)

MO TREATMENT OF HETERONUCLEAR DIATOMIC MOLECULES

The treatment is similar to that for homonuclear diatomic molecules.

Ali Ebrahimi ۴۴ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

Cross section of a contour Cross section of the πu2p+1 (or of the 1π±1 MOs in CO, πu2p-1) molecular orbital. determined by an extended-basis-set SCF calculation

The ground-state configuration:

CO:

N2:

MOs are approximated as LCAOs. The coefficients are found by solving the Roothaan equations

By a minimal-basis-set SCF calculation using Slater AOs (with nonoptimized exponents given by Slater's rules) for CO:

π MOs are simpler than σ MOs

Ali Ebrahimi ۴۵ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

corresponding MOs in N2 at R = Re:

1σ MO in CO  1s oxygen-atom AO 2σ MO in CO  1s -atom AO.

for AB, where the valence AOs :  are of s and p type  of A do not differ greatly in energy from the valence AOs of B

Homonuclear diatomic Heteronuclear diatomic

each valence AO of the more electronegative atom would lie below the corresponding valence AO of the other atom.

Ali Ebrahimi ۴۶ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

When the s and pσ valence AOs of B lie below the s valence-shell AO of A: this affects which AOs contribute to each MO

the molecule BF by a minimal-basis-set calculation:

 1σ  1sF  2σ  1sB  3σ  predominantly 2sF, with small amounts of 2sB, 2pσB, and 2pσF.  4σ  predominantly 2pσF, with significant amounts of 2sB and 2sF and a small amount of 2pσB  1π  a bonding combination of 2pπB and 2pπF  5σ  predominantly 2sB, with a substantial contribution from 2pσB and a significant contribution from 2pσF  2π  an antibonding combination of 2pπB and 2πpF.  6σ  important contributions from 2pσB, 2sB, 2sF, and 2pσF.

2pF AO lies well below the 2sB AO, thus:  2pσF AO contribute substantially to lower-lying MOs  2sB AO contribute substantially to higher-lying MOs

CO:

4σ : very substantial contribution from 2pσO 5σ : very substantial contribution from 2sC

For AB where each atom has s and p valence-shell AOs and the A and B valence AOs differ widely in energy

but it is not so easy to guess which AOs contribute to the various MOs or the bonding or antibonding character of the MOs.

Ali Ebrahimi ۴٧ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

Diatomic hydrides: H has only a 1s valence AO

HF: 1) a crude qualitative approximation ground-state configurations, Is for H and 1s22s22p5 for F 1s and 2s F : take little part in the bonding. 2pπ F electrons are nonbonding Is AO of H and 2pσ AO of F have the same symmetry (σ) and similar energies, and a linear combination of these two AOs will form a σ MO for the bonding electron pair:

F is more electronegative than H, we expect that c2> c1.

2) A minimal- basis-set SCF calculation using Slater orbitals with optimized exponents:

The ground-state configuration of HF : 1σ22σ23σ21π4

Since a single 2s function is only an approximation to the 2s AO of F, we cannot use this calculation to say exactly how much 2s AO character the 3σ HF molecular orbital has.

Ali Ebrahimi ۴٨ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

heteronuclear diatomic molecules:

Accurate MO expressions : the solution of the Roothaan equations In the crudest approximation: valence MO ≈ a linear combination of two AOs

two Aos, one on each atom

coefficients are not equal (lack of symmetry) The coefficients are determined by solving the secular equation

f(w) Ĥ is some sort of effective one-electron Hamiltonian

Suppose that Haa > Hbb,

Sab is less than 1 (except at R = 0). 2 The coefficient of W in f(W) is (1 - Sab) > 0  f() = f(-) = + > 0.

W = Haa or Hbb   f(Haa) < 0 and f(Hbb) < 0.

zero The roots occur where f(w) equals 0

Therefore, the orbital energy of one MO is less than both Haa and Hbb and the energy of the other MO is greater than both Haa and Hbb. F(w)

w

w1 w2

Haa Hbb

Ali Ebrahimi ۴٩ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

Formation of bonding and antibonding MOs from AOs in the homonuclear and heteronuclear cases.

These figures are gross oversimplifications, since a given MO has contributions from many AOs

VB TREATMENT OF HETERONUCLEAR DIATOMIC MOLECULES

valence-bond ground state wave function of HF: The Heitler-London function corresponding to pairing hydrogen 1s electron and 2pσ electron.

… ≡

essentially covalent

An ionic valence-bond function has the form a(1)a(2)

The VB wave function is then written as

Ali Ebrahimi ۵٠ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

Cs: the lowest ionization potential, 3.9 eV. Cl: the highest ea, 3.6 eV. F: ea = 3.45 eV CsCl  Cs + Cl CsF  Cs + F There are cases of excited states of diatomic molecules that dissociate to ions.

Ali Ebrahimi ۵١ ١۴۴٠/٠٢/٠۵ Univ. of Sistsn and Baluchestan

THE VALENCE-ELECTRON APPROXIMATION

Cs2, which has 110 electrons In the MO method:

110 X 110 Slater determinant of molecular orbitals

MOs  functions containing variational parameters

minimize the variational integral

the valence-electron approximation:

108 core electrons + two 6s valence electrons molecular energy = core- and valence-electron energies.

1) core electrons ≡ point charges coinciding with the nucleus.

Hamiltonian (for Cs2) = Hamiltonian for H2

minimize the variational integral no restrictions on the valence-electron trial functions,

valence-electrons' MO to "collapse" to the σg1s MO

Constraint: variational functions used for the valence electrons be orthogonal to the orbitals of the core electrons.

more work 2) core electrons are treated as a charge distribution (effective repulsive potential for the motion of the valence electrons). effective Hamiltonian for the valence electrons The valence-electron approximation is widely used in approximate treatments of polyatomic molecules

Ali Ebrahimi ۵٢